In quantum mechanics, a wave function is a mathematical description of the quantum state of an isolated quantum system. It is represented by the Greek letter Ψ and is a function of the degrees of freedom corresponding to some maximal set of commuting observables. The wave function describes the position and state of a particle, and its square gives the probability density of finding the particle described by a specific wave function Ψ at a given point and time. The wave function is a complex-valued probability amplitude in the Copenhagen interpretation of quantum mechanics. The Schrödinger equation determines how wave functions evolve over time, and a wave function behaves qualitatively like other waves, such as water waves or waves on a string, because the Schrödinger equation is mathematically a type of wave equation. The wave function is not unique to quantum mechanics and is used in many other systems, such as the motion of ripples of water, sound waves, vibrations on a string, or electromagnetic waves.

The properties and postulates of the wave function include:

• The wave function is a mathematical description of a quantum state of a particle as a function of momentum, time, position, and spin.
• The symbol used for a wave function is a Greek letter called psi, Ψ.
• The probability distribution in three dimensions is established using the wave function.
• The probability of finding a particle if it exists is 1.
• For a particle in a conservative field of force system, using the wave function, it becomes easy to understand the system.
• For every physical observable q, there is an operator Q operating on a wave function associated with a definite value of that observable such that it yields a wave function of that many times.

In summary, a wave function is a mathematical description of the quantum state of an isolated quantum system, and it describes the position and state of a particle. Its square gives the probability density of finding the particle described by a specific wave function Ψ at a given point and time. The wave function is a complex-valued probability amplitude, and it behaves qualitatively like other waves. The Schrödinger equation determines how wave functions evolve over time.